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Results tagged with set-theory
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user 22131
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
6
votes
Which models of set theory are locally presentable?
Your category of fake set is going to be in particular a boolean elementary topos. By Giraud's theorem, It is locally presentable if and only if it is a Grothendieck topos, so a category of sheaves ov …
- 42.4k
27
votes
Does Zorn's Lemma imply a physical prediction?
There are a lot of argument that can be applied here, and the question linked in the comment already give several of these, but there is one that I really like, and which I don't remember having seen …
- 42.4k
21
votes
Accepted
There are no abstract categories
The argument you give in your original post is essentially the proof that every small category is concrete.
So if you work in a setting that have enough universes/inaccessible cardinal/notion of small …
- 42.4k
10
votes
Accepted
Remark 5.4.2.15 in HTT
He is applying 5.4.2.13 to $C$ and not to $D$: Because $C^\kappa$ is essentially small, and each $F(c)$ is $\lambda_c$-compact for some $\lambda$; there is a $\kappa = \sup_{c \in C^\kappa} \lambda_c$ …
- 42.4k
26
votes
Accepted
Why do we care about small sets?
First, it is important to distinguish between the problem related to the foundation you are using from the problems that are inherent to category theory.
For example, the distinction between $\mathbb{ …
- 42.4k
11
votes
2
answers
661
views
Non smallness of the set of anafunctors without AC?
Trying to construct a model category constructively is difficult. One often mention the fact that without the axiom of choice one cannot prove that the localization of the category of small categories …
- 42.4k
21
votes
Accepted
Complete Boolean algebra not isomorphic to a $\sigma$-algebra
Let $\Sigma_0$ be the $\sigma$-algebra of Lebesgue measurable sets on the real interval $[0,1]$. Define $\Sigma$ to be $\Sigma_0$ quotiented by the relation $U \simeq V$ if $U$ and $V$ differ by a Leb …
- 42.4k
21
votes
Accepted
Are there substantive differences between the different approaches to "size issues" in categ...
From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is me …
- 42.4k
1
vote
Set-theoretic forcing over sites?
Are you familiar with this paper : Relating first order set theories and elementary toposes from Awodey,Butz,Simpson and Streicher. I haven't read in detail yet, but it really seems to provide a machi …
- 42.4k
11
votes
Is material set theory conservative over structural set theory?
This will obviously be highly dependent on the concrete theory you are considering. But overall the answer is yes.
The most general version of these results I'm aware of are in Mike Shulman's Comparin …
- 42.4k
13
votes
Accepted
Characterization of Stone-Cech compactifications
I confirme my comment :
$X$ is the stone-cech compactification of a discrete space if and only if $X$ is compact, haussdorf, extremally disconected, and has a dense set of open points.
here is a ske …
- 42.4k
2
votes
Is the Jordan decomposition of a self-adjoint functional constructive?
Let me expand a little my comment because it is more subbtle than what I suggested and it relies heavily on a recent result. I agree that this does not exactly answser your question but is I think rel …
- 42.4k
7
votes
Is there any danger far from home? (Edited & Revised Version)
If you don't allow in your theory (or your logic) axioms or rules that allow to use an infinite number of proposition to deduce a new one (which is not possible if you stick to "finitary logic" ) then …
- 42.4k
5
votes
Accepted
Failure of SVC in Grothendieck toposes
It seem to me that the problem that Makkai has in mind is that the existence of non-trivial choice objects is in conflict with non-booleaness.
The core of the arguement, is the following lemma, whic …
- 42.4k
4
votes
Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / mul...
So, I wanted to say something, I don't think this answer the questions, but that was definitely too long for a comment. In short, this is just the result of me trying to make sense of this idea:
Comin …
- 42.4k